Numbers and Percentages

Similar to Cause and Effect Reasoning, Conditional Reasoning, and Formal
Logic, the concept of Numbers and Percentages is featured in many LSAT
stimuli. Although most people are comfortable working with numbers or
percentages because they come up so frequently in daily life (for example in
balancing a checking account, dividing a bar tab, or adding up a grocery bill),
the makers of the LSAT often prey upon several widely-held misconceptions:
Misconception #1: Increasing percentages automatically lead to
increasing numbers.
Most people assume that if a percentage becomes larger, the number that
corresponds to that percentage must also get larger. This is not
necessarily true because the overall size of the group under discussion
could get smaller. For example, consider the following argument: “Auto
manufacturer X increased their United States market share from 10%
last year to 25% this year. Therefore, Company X sold more cars in the
United States this year than last.” This is true if the size of the U.S. car
market stayed the same or became larger. But if the size of the U.S. car
market decreased by enough, the argument would not be true, as in the
following example:
                              Last Year            This Year
Total number of cars                   1000              200
sold in the United States
X’s market share                       10%               25%
X’s total car sales                      100               50
in the United States
Thus, even though auto manufacturer X’s market share increased to
25%, because the size of the entire market decreased significantly, X
actually sold fewer cars in the United States.
Misconception #2: Decreasing percentages automatically lead to
decreasing numbers.
This misconception is the opposite of Misconception #1. Just because a
percentage decreases does not necessarily mean that the corresponding
number must become smaller. Reversing the years in the previous
example proves this point.
Misconception #3: Increasing numbers automatically lead to increasing
percentages.
Just as increasing percentages do not automatically translate into
increasing numbers, the reverse is also true. Consider the following
example: “The number of bicycle-related accidents rose dramatically
from last month to this month. Therefore, bicycle-related accidents must
make up a greater percentage of all road accidents this month.” This
conclusion can be true, but it does not have to be true, as shown by the
following example:
                        Last Month          This Month
Number of bicycle-related            10              30
accidents
Total number of road accidents        100             600
Percentage of total accidents          10%             5%
that are bicycle-related
Thus, even though the number of bicycle-related accidents tripled, the
percentage of total road accidents that were bicycle-related dropped
because the total number of road accidents rose so dramatically.
Misconception #4: Decreasing numbers automatically lead to decreasing
percentages.
This misconception is the opposite of Misconception #3. Just because a
number decreases does not necessarily mean that the corresponding
percentage must become smaller. Reversing the months in the previous
example proves this point.
Misconception #5: Large numbers automatically mean large
percentages, and small numbers automatically mean
small percentages.
In 2003, Porsche sold just over 18,000 cars in the United States. While
18,000 is certainly a large number, it represented only about 1/5 of 1%
of total U.S. car sales in 2003. Remember, the size of a number does not
reveal anything about the percentage that number represents unless you
know something about the size of the overall total that number is drawn
from.
Misconception #6: Large percentages automatically mean large
numbers, and small percentages automatically mean
small numbers.
This misconception is the reverse of Misconception #5. A figure such as
90% sounds impressively large, but if you have 90% of $5, that really
isn’t too impressive, is it?
Numerical situations normally hinge on three elements: an overall total, a
number within that total, and a percentage within the total. LSAT problems will
often give you one of the elements, but without at least two elements present,
you cannot make a definitive judgment about what is occurring with another
element. When you are given just percentage information, you cannot make a
judgment about numbers. Likewise, when you are given just numerical
information you cannot make a judgement about percentages.
In a moment, we will explore this idea by examining several LSAT questions.
But first, you must be able to recognize number and percentage ideas when they
appear on the LSAT:
Words used to introduce numerical ideas:
Amount
Quantity
Sum
Total
Count
Tally
Words used to introduce percentage ideas:
Percent
Proportion
Fraction
Ratio
Incidence
Likelihood
Probability
Segment
Share
Three words on the percentage list—“incidence, “likelihood,” and
“probability”—bear further discussion. Each of these words relates to the
chances that an event will occur, and when the LSAT makers uses phrases such
as “more likely” or “less likely” they are telling you that the percentage chances
are greater than 50% or less than 50%, respectively. In fact, a wide variety of
phrases can be used to introduce percentage ideas, including such disparate
phrases as “more prone to” or “occurs with a high frequency.”
With these indicators in mind, please take a moment to complete the following
question:
1. From 1973 to 1989 total energy use in this country increased less than 10 percent. However, the use of energy in this country during this same period grew by more than 50 percent, as did the gross national product—the total value of all goods and services produced in the nation.
If the statements above are true, then which one of
the following must also be true?
(A) Most of the energy used in this country in
1989 was electrical energy.
(B) From 1973 to 1989 there was a decline in the
use of energy other than electrical energy in
this country.
(C) From 1973 to 1989 there was an increase in
the proportion of energy use in this country
that consisted of electrical energy use.
(D) In 1989 electrical energy constituted a larger
proportion of the energy used to produce the
gross national product than did any other
form of energy.
(E) In 1973 the electrical energy that was
produced constituted a smaller proportion of
the gross national product than did all other
forms of energy combined.
Like the vast majority of Must Be True problems, the stimulus does not contain
a conclusion. We are given the following facts, however:
From 1973 to 1989 total energy use increased less than 10%.
During this same period, the use of electrical energy grew by more than
50%.
During this same period, the gross national product (GNP) grew by
more than 50%.
A careful examination of the second sentence reveals that there is no stated
connection between the growth of the GNP and the increase in the use of
electrical energy. If you assume that the use of electrical energy somehow
caused the growth of the GNP, you are guilty of making an unwarranted causal
assumption. Because there is no stated connection between the two other than
they both grew by more than 50%, any answer that attempts to connect the two
is incorrect. Answer choices (D) and (E) can both be eliminated by this
reasoning.
Now that we recognize that the GNP issue is only a red herring, let us examine
the percentages that are given in the stimulus. The 50% increase in electrical
energy gives the impression that the jump must have been substantial. But we
know from Misconception #6 that a large percentage does not automatically
mean a large number. For example, in this problem it is possible that the 50%
increase in electrical energy use was a jump from 2 units to 3 units. The
possibility that electrical energy use in 1973 was a relatively small percentage of
overall energy use directly undermines answer choices (A), as shown by the
following example:
                          1973               1989
Total energy use                 100              109
(in units)
Electrical energy use                10              15
(in units)
Percentage of total energy             10%           13+%
use that was electrical
A close analysis of the chart also reveals that answer choice (B) can be
eliminated. In the example, the use of energy other than electrical energy rose
from 90 units to 94 units.
Although the example disproves both answer choice (A) and (B), obviously
you do not have time to make a chart during the test to examine each possibility,
so is there a faster way to eliminate the first two answers? Yes—consider the
previous discussion point that information about percentages does not tell us
about the numbers. With that idea in mind, because the stimulus contains only
percentage information (even though there are two percentages), you should be
very suspicious of answer choice (A) (which states that the number of electrical
units used was greater) and answer choice (B) (which states that the use of nonelectrical
energy declined) since they both contain numerical information. At the
same time, you should be attracted to an answer such as (C) because it contains
only percentage information, and as it turns out, answer choice (C) is correct.
Because the misconceptions discussed earlier have a predictable effect when
you try to make inferences, you can use the following general rules for Must Be
True questions:
1. If the stimulus contains percentage or proportion information only, avoid
answers that contain hard numbers.
Example Stimulus Sentence:
The car market share of Company X declined this year.
Avoid answers which say:
Company X sold a smaller number of cars this year.
Company X sold a greater amount of cars this year.
2. If the stimulus contains only numerical information, avoid answers that
contain percentage or proportion information.
Example Stimulus Sentence:
Company Y sold fewer computers this year.
Avoid answers which say:
Company Y now has a lower share of the computer market.
Company Y now possesses a greater proportion of the
computer market.
3. If the stimulus contains both percentage and numerical information, any
answer choice that contains numbers, percentages, or both may be true.
Please keep in mind that these rules are very general. You must read the
stimulus closely and carefully to determine exactly what information is present
because the makers of the LSAT are experts at camouflaging or obscuring
important information in order to test your ability to understand complex
argumentation.
Please take a moment to complete the following question:
2. The number of North American children who are
obese—that is, who have more body fat than do 85
percent of North American children their age—is
steadily increasing, according to four major studies
conducted over the past 15 years.
If the finding reported above is correct, it can be
properly concluded that
(A) when four major studies all produce similar
results, those studies must be accurate
(B) North American children have been
progressively less physically active over the
past 15 years
(C) the number of North American children who
are not obese increased over the past 15
years
(D) over the past 15 years, the number of North
American children who are underweight has
declined
(E) the incidence of obesity in North American
children tends to increase as the children
grow older
Like the previous question, this is a Must Be True question with a stimulus that
does not contain a conclusion. But, this stimulus does provide information about
both the numbers and percentages of obese children, and so you can end up
with an answer that has either a number or a percentage (though a numerical
answer is more likely since the percentage is fixed at a constant 15% in the
stimulus).

The numerical information comes from the phrase, “The number of North
American children who are obese...is steadily increasing.” The percentage
information comes from the phrase, “children who are obese—that is, who have
more body fat than do 85 percent of North American children their age.” The
percentage information defines obese children as those who fall into the top
15% among all children their age in terms of body fat, and therefore the
percentage is known to be constant. The numerical information tells us that the
actual number of obese children is increasing (and since this is a Must Be True
question we can accept that information as accurate).
Answer choice (A): This answer is incorrect because there is no evidence in the
stimulus to support it. Although the stimulus mentioned four major studies that
apparently agreed about the increase in the number of obese children, it would
be an exaggeration to say that any time four major studies produce similar
results they must be accurate.
Answer choice (B): This answer proposes a causal reason for why the number
of obese children is growing. From the information in the stimulus we cannot
determine the cause of the rise in obesity, so answer choice (B) is also wrong.
Answer choice (C): This is the correct answer. Consider the following example:
15 years ago—100 total children of similar age
Number of obese children 15 = 15%
Number of non-obese children 85
Now, let us say that the number of obese children has risen to 150 children
today:
Today
Number of obese children 150
So far we have conformed to the information given in the stimulus: the actual
number of obese children is rising. However, although the number of obese
children has now risen to 150, the definition of obesity (“more body fat than 85
percent of North American children”) remains unchanged. Since this is the case,
the 150 obese children today must still comprise the top 15% of the total child
population. Consequently, the remaining 85% of non-obese children must now
be 850:
Today
Number of non-obese children 850
(150 is 15% of 1000, and thus 85% of 1000 is 850)
Answer choice (C) is fully supported because the stimulus provides information
about both the number and percentage of obese children. As stated earlier, if the
stimulus provides information about both the numbers and percentages in a
situation, then you can select any supported answer choice that contains either
numbers or percentages. Note the emphasis on the word “supported.” In the
obesity problem, Law Services could easily have written an incorrect answer
choice that says, “The number of North American children who are not obese
decreased over the past 15 years.”
Answer choice (D): This answer addresses “underweight” children, who are
neither defined nor discussed in the stimulus.
Answer choice (E): This answer is directly contradicted by the information in
the stimulus, which states that the incidence of obesity is definitionally set at a
constant 15%.
Both of the previous questions were Must Be True questions, but of course the
makers of the LSAT can also ask other questions about a stimulus that contains
numbers and percentages. Please take a moment to consider the following
problem:
3. Waste management companies, which collect waste
for disposal in landfills and incineration plants,
report that disposable plastics make up an
ever-increasing percentage of the waste they handle.
It is clear that attempts to decrease the amount of
plastic that people throw away in the garbage are
failing.
Which one of the following, if true, most seriously
weakens the argument?
(A) Because plastics create harmful pollutants
when burned, an increasing percentage of
the plastics handled by waste management
companies are being disposed of in landfills.
(B) Although many plastics are recyclable, most of
the plastics disposed of by waste
management companies are not.
(C) People are more likely to save and reuse
plastic containers than containers made of
heavier materials like glass or metal.
(D) An increasing proportion of the paper, glass,
and metal cans that waste management
companies used to handle is now being
recycled.
(E) While the percentage of products using plastic
packaging is increasing, the total amount of
plastic being manufactured has remained
unchanged.
The structure of the argument, in simplified form, is as follows:
Premise: Disposable plastics make up an ever-increasing percentage
of the waste they handle.
Conclusion: Attempts to decrease the amount of plastic that people throw
away in the garbage are failing.
Based on our discussion of numbers and percentages, it should be clear that the
conclusion is flawed: a numbers conclusion (“amount”) cannot be drawn solely
from percentage information because the overall total could change
dramatically. As you attack the answer choices, look for an answer that
addresses this error.
Answer choice (A): The argument is about how people act when throwing
away garbage, an issue that occurs before the waste management companies
receive the trash. On the other hand, this answer discusses how the waste
management companies dispose of plastics, an issue that occurs after they have
received the waste. Because the two issues occur at different times in the cycle,
this answer does not attack the argument and is incorrect.
Answer choice (B): Like answer choice (A), this answer raises an issue that
occurs after the waste management companies have received the waste.
Answer choice (C): This answer addresses how people act prior to throwing
away garbage, but it does not suggest that the amount of plastic that people
throw away is not decreasing. The author would probably counter this
statement by saying that regardless of the fact that people are more likely to save
plastic containers, that tendency is only relative to glass and metal containers,
and people are still throwing away plastics in an ever-increasing percentage
(and thus amount).
Answer choice (D): This is the correct answer. The answer indicates that the
waste management companies no longer receive as much paper, glass, and
metal as they used to. Since this clearly affects the amount of trash that they
process, this would also affect the percentages of each type of waste. If the
amount of paper, glass, and metal drops by a large amount, the percentage of
plastic in the waste would rise even if the actual amount of plastic waste was
reduced. The following example shows how this is possible:
                          Previously             Now
Total garbage                      100              20
(in units)
Plastic garbage                      20 (20%)        10 (50%)
(in units)
Other garbage                         80 (80%)       10 (50%)
(in units)
In the example, plastic garbage has risen from 20% to 50%, but the actual
amount of plastic waste has decreased from 20 units to 10 units. Consequently,
because this answer raises a scenario that could disprove the argument, it is the
correct answer.
Answer choice (E): The amount of plastic being manufactured is not the issue
in the stimulus; how much plastic is thrown away is the issue.
In all respects this is a classic numbers and percentages Weaken problem.
Accordingly, we can use this discussion to highlight a general rule for handling
Weaken and Strengthen questions paired with numbers and percentages stimuli:
To weaken or strengthen an argument containing numbers and
percentages, look carefully for information about the total amount(s)—
does the argument make an assumption based on one of the
misconceptions discussed earlier?
On the following page, another numbers and percentage problem is presented.
Please take a moment to complete the following question.
4. For next year, the Chefs’ Union has requested a 10
percent salary increase for each of its members,
whereas the Hotel Managers’ Union has requested
only an 8 percent salary increase for each of its
members. These facts demonstrate that the average
dollar amount of the raises that the Chefs’ Union has
requested for next year is greater than that of the
raises requested by the Hotel Managers’ Union.
Which one of the following, if true, most strengthens
the argument?
(A) The Chefs’ Union has many more members than
does the Hotel Managers’ Union.
(B) The Chefs’ Union is a more powerful union
than is the Hotel Managers’ Union and is
therefore more likely to obtain the salary
increases it requests.
(C) The current salaries of the members of the
Chefs’ Union are, on average, higher than the
current salaries of the members of the Hotel
Managers’ Union.
(D) The average dollar amount of the raises that the
members of the Chefs’ Union received last
year was equal to the average dollar amount
of the raises that the members of the Hotel
Managers’ Union received.
(E) The members of the Chefs’ Union received
salary increases of 10 percent in each of the
last two years, while the members of the
Hotel Managers’ Union received salary
increases of only 8 percent in each of the last
two years.
This problem makes the classic mistake of assuming that a larger percentage
translates into a greater number (Misconception #6). According to the argument,
because the Chef’s Union requested a 10% raise and the Hotel Manager’s
Union requested only an 8% raise, the Chef’s Union must have asked for more
money than the Hotel Manager’s Union. But, the argument never tells us how
much the average member of each union makes, so the conclusion cannot be
drawn with certainty, as shown by the following example:
                       Chef             Hotel
Raise request                10%            8%
Average current salary          $1000          $10,000
Actual amount of raise          $100           $800
requested
Even though the Chef’s Union has asked for a greater percentage raise than the
Hotel Manager’s Union, it is still possible that the actual dollar amount of the
Hotel Manager’s Union request is greater. In this case, omitting the average
current salary made by each member is tantamount to omitting the total amount
made by the members, and thus, even though this problem uses averages, it
trades on the mistake behind all the misconceptions discussed at the beginning
of this chapter. To strengthen the argument, you must find an answer that
indicates that the Chef’s Union has a wage that is equal to or greater than the
wage of the Hotel Manager’s Union (the wage could also be very slightly
below that of the Hotel Manager’s Union).
Answer choice (A): Because the conclusion is specific about the average dollar
amount requested, and an average can be calculated regardless of how many
members are in the union, this answer is irrelevant to the argument.
Answer choice (B): The argument focuses on the size of each Union’s raise
request. Whether each union will receive the request is not at issue, and thus this
answer is incorrect.
Answer choice (C): This is the correct answer. As discussed above, an answer
that indicates that the Chef’s Union has a wage that is equal to or greater than
the wage of the Hotel Manager’s Union would strengthen the argument. This is
the answer you should look for when you read the question stem, and you
should attempt to accelerate through the answer choices to find this answer.
Answer choice (D): This answer refers to the raises given out last year.
Unfortunately, this fails to address the current salaries of the union members.
Answer choice (E): Like answer choice (D), this answer addresses previous
raises, which does not tell us about current salaries.